
\(c_3-c_2, c_5 \ne 0\) : The constant term on the right hand side just shifts the boundary potential by \(\textrm {const}/c_5\) , which has no physical consequences. Therefore, it suffices to take this constant to be zero, and the solution reads
\(\theta (\phi ) = \mathfrak {c}_1 \cosh \left( \mathfrak {c}_2 \phi - \phi _0 \right), \quad \mathfrak {c}_2 = \sqrt{\frac{c_5}{c_3-c_2}},\) 
where \(\mathfrak {c}_1, \phi _0\)  denote constants of integration. This result essentially represents the boundary potential for the Sine-Gordon theory as found by Ghoshal and Zamolodchikov [1]} .

\(c_5 =0, c_3-c_2 \ne 0\) : The constant term on right hand side is now important, since it yields a quadratic contribution to the potential:
\(\theta (\phi )= \frac{\textrm {const}}{2(c_3-c_2)} \phi ^2 + \mathfrak {c}_3 + \mathfrak {c}_4 \phi \, ,\) 
where \(\mathfrak {c}_3, \mathfrak {c}_4\)  denote constants of integration. Physically they are not important, since we can absorb them by a constant field shift. This is the boundary function discussed in [2]}.

\(c_3 -c_2 =0, c_5 \ne 0\)  case: the boundary potential is a constant. Consequently, the bulk field satisfies Neumann boundary conditions.

